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Fundam. Prikl. Mat., 2000, Volume 6, Issue 3, Pages 649–668 (Mi fpm496)  

This article is cited in 1 scientific paper (total in 1 paper)

Exponential Diophantine equations in rings of positive characteristic

A. Ya. Belova, A. A. Chilikovb

a House of scientific and technical work of youth
b M. V. Lomonosov Moscow State University

Abstract: In this work we prove the algorithmical solvability of the exponential-Diophan-tine equations in rings represented by matrices over fields of positive characteristic. Consider the system of exponential-Diophantine equations
$$ \sum_{i=1}^{s}P_{ij}(n_1,\ldots,n_t)b_{ij0}a_{ij1}^{n_1}b_{ij1}\ldots a_{ijt}^{n_t}b_{ijt}=0 $$
where $b_{ijk},a_{ijk}$ are constants from matrix ring of characteristic $p$, $n_i$ are indeterminates. For any solution $\langle n_1,\ldots,n_t \rangle$ of the system we construct the word (over alphabet which contains $p^t$ symbols) $\overline\alpha_0\ldots\overline\alpha_q$, where $\overline\alpha_i$ is a $t$-tuple $\langle n_1^{(i)},\ldots,n_t^{(i)}\rangle$, $n^{(i)}$ is the $i$-th digit in the $p$-adic representation of $n$. The main result of this work is: the set of words, corresponding in this sense to the solutions of the system of exponential-Diophantine equations is a regular language (i. e. recognizible by a finite automaton). There is an effective algorithm which calculates this language.

Full text: PDF file (799 kB)

Bibliographic databases:

UDC: 512.5+511
Received: 01.03.1998

Citation: A. Ya. Belov, A. A. Chilikov, “Exponential Diophantine equations in rings of positive characteristic”, Fundam. Prikl. Mat., 6:3 (2000), 649–668

Citation in format AMSBIB
\Bibitem{BelChi00}
\by A.~Ya.~Belov, A.~A.~Chilikov
\paper Exponential Diophantine equations in rings of positive characteristic
\jour Fundam. Prikl. Mat.
\yr 2000
\vol 6
\issue 3
\pages 649--668
\mathnet{http://mi.mathnet.ru/fpm496}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1801320}
\zmath{https://zbmath.org/?q=an:0990.11079}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Kanel-Belov, A, “Normal bases of PI-algebras”, Advances in Applied Mathematics, 37:3 (2006), 378  crossref  mathscinet  zmath  isi
  • Фундаментальная и прикладная математика
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